Derivatives of the following...

Question

Find d^2/dx^2

1. sinx + x^2y = 7; (2,1)

2. cos(x-y)^2 + siny = sqrt(2); ((pi/2),(pi/4))

3. (x^2 + y^2)^2 + 25/4xy^2; (1,2)

Answer 1

These derivatives require the use of implicit differentiation (basically just an application of the chain rule. If you could solve these equations for y, you would have y = some function of x and you would use the regular rules of differentiation you have learned (power rule, product rule, quotient rule, etc). But you can't solve them for y, at least not easily. So we use implicit differentiation. When you get to "y" in the equation, you take the derivative of y then multiply by dy/dx (as the chain rule dictates.

Example: xy = 2

x and y are being multiplied so that means you would use the product rule. So the derivative of xy is:

x'y + y'x

x' = 1 and y' = dy/dx

So the derivative is 1•y + dy/dx•x

Then the derivative of "2" is zero so the equation becomes:

1•y + dy/dx•x = 0

Solving for dy/dx we get dy/dx = -y/x

In this particular case, it is in fact easy to solve for y. y = 2/x so plugging in "2/x" in for "y" in dy/dx = -y/x gives dy/dx = -(2/x)/x = -2/x² and that is the same result you would get if you took the derivative of y = 2/x using the power rule or the quotient rule.

The same process is used on the problems you are asking for help on. However, I'm a bit confused about what the problems are asking because it looks like there is a point after the equation however that point is not on the function.

Using this process, you should get (for problem 1) a second derivative of (6xy + xsin(x) + 4cos(x))/x³

If you need help walking you through this, let me know and I'll provide a free tutoring session to discuss it.